This paper introduces a new extragradient-type algorithm for a class of nonconvex-nonconcave minimax problems. It is well-known that finding a local solution for general minimax problems is computationally intractable. This observation has recently motivated the study of structures sufficient for convergence of first order methods in the more general setting of variational inequalities when the so-called weak Minty variational inequality (MVI) holds. This problem class captures non-trivial structures as we demonstrate with examples, for which a large family of existing algorithms provably converge to limit cycles. Our results require a less restrictive parameter range in the weak MVI compared to what is previously known, thus extending the applicability of our scheme. The proposed algorithm is applicable to constrained and regularized problems, and involves an adaptive stepsize allowing for potentially larger stepsizes. Our scheme also converges globally even in settings where the underlying operator exhibits limit cycles.

翻译：本文针对一类非凸-非凹极小极大问题提出了一种新的外梯度型算法。众所周知，求解一般极小极大问题的局部解在计算上是棘手的。这一观察近期推动了在变分不等式更一般框架下，当所谓的弱Minty变分不等式成立时，一阶方法收敛性所需结构的研究。我们通过示例展示了该类问题具有非平凡的结构，而现有大量算法在此类问题中已被证明会收敛至极限环。与先前研究相比，我们的结果要求弱MVI中参数范围更宽松，从而拓展了所提方案的适用性。该算法适用于约束和正则化问题，并采用自适应步长以允许可能更大的步长。即使底层算子存在极限环，我们的方案也能实现全局收敛。